Let \(H_{1}\in \mathcal {K}_{1}\) . Then \(H_{1}\)  is \((C,\ell )\) -elation group of
\(PG_{2}(3)\)  and hence \(N_{PSL_{3}(3)}(H_{1})\)  is the stabilizer in \(PSL_{3}(3)\)  of the flag \((C,\ell )\) . Thus \(N_{PSL_{3}(3)}(H_{1})\cong U:E_{4} \) , and hence \(N_{PSL_{3}(3):\left\langle \sigma \right\rangle }(H_{1})=U:D_{8}\) , where \(U\)  is a Sylow 3-subgroup of \(PSL_{3}(3)\) , since \(\mathcal {K}_{1}\)  and \(\mathcal {K}_{2}\) , are conjugacy \(PSL_{3}(3):\left\langle \sigma \right\rangle \) -classes. This proves (1).
Let \(H_{2}\in \mathcal {K}_{2}\)  and let \(\mathcal {C}\)  be any non-degenerate
conic of \(PG_{2}(3)\) . It is well known that \(PSL_{3}(3)_{\mathcal {C}}\cong PGL_{2}(3)\cong S_{4}\)  and that \(PSL_{3}(3)_{\mathcal {C}}\)  does not
contain elations of \(PG_{2}(3)\)  (e.g. see [1]}, Theorem 2.37). Then any
Sylow 3-subgroup of \(PSL_{3}(3)_{\mathcal {C}}\)  lie in \(\mathcal {K}_{2}\) .
Hence, we may assume that \(H_{2}\)  is such a group. Then \(S_{3}\leqslant N_{PGL_{2}(3)}(H_{2})\)  and hence \(N_{PGL_{2}(3)}(H_{2})\cong E_{3}\times S_{3}\) , since \(C_{PSL_{3}(3)}(H_{2})\cong E_{9}\)  by [2]}. Now, it is
easy to see that \(N_{PGL_{2}(3):\left\langle \sigma \right\rangle }(H_{2})\cong S_{3}\times S_{3}\) , which is (2).
